The Cauchy problem for a fourth-order thin film equation

Abstract

In this talk we will discuss different aspects of the Cauchy problem of a class of fourth order thin film equation of the form $u_t=\Delta \cdot (\left(|u{|}^n\nabla \Delta u\right)$. We will show recent results in which we obtained a countable number of similarity solutions of the thin film equation via a homotopy transformation as $n\to 0+$ to the similarity solutions of the classic bi-harmonic equation $u_t={\Delta }^{2}u$. Also, another similar homotopic approach is performed directly from the thin film equation to the parabolic bi-harmonic equation in order to obtain important properties for the Cauchy problem. This is a joint work with Prof. Victor. A. Galaktionov at the University of Bath (UK).